Asymptotic behaviour of pion-pion total cross-sections

Abstract

We derive a sum rule which shows that the Froissart-Martin bound for the asymptotic behaviour of the ππ total cross sections at high energies, if modulated by the Lukaszuk-Martin coefficient of the leading log 2 s behaviour, cannot be an optimal bound in QCD. We next compute the total cross sections for π + π − , π ± π 0 and π 0 π 0 scattering within the framework of the constituent chiral quark model (CχQM) in the limit of a large number of colours N c and discuss their asymptotic behaviours. The same ππ cross sections are also discussed within the general framework of Large-N c QCD and we show that it is possible to make an Ansatz for the isospin I = 1 and I = 0 spectrum which satisfy the FroissartMartin bound with coefficients which, contrary to the Lukaszuk-Martin coefficient, are not singular in the chiral limit and have the correct Large-N c counting. We finally propose a simple phenomenological model which matches the low energy behaviours of the σ total π ± π 0 (s) cross section predicted by the CχQM with the high energy behaviour predicted by the Large-N c Ansatz. The magnitude of these cross sections at very high energies is of the order of those observed for the pp and pp scattering total cross sections.

Can Froissart bound explain hadron cross-sections at high energies?

“…Since large impact parameters correspond to a small invariant mass of the pion pair, and low energy Goldstone bosons decouple, it was argued in [9,10] that σ 0 would be determined by the strong interaction scale of M ∼ 1 GeV, σ 0 ∼ 1/M 2 . A similar conclusion was also reached in [11,12] and in [13,14], where σ 0 was calculated to be given by some combination of ρ-meson mass and f π .…”

Froissart bound for/from CFT Mellin amplitudes

“…where the last inequality was possible because n ≥ 0. Thus we have the following inequality, 8 In footnote 6 it was mentioned that, the analysis so far was carried out by implicitly considering ∆ φ ∼ τ along with ∆ φ 1, τ 1. However, observe that, the upper limit of the τ−sum really does not conform to τ ∼ ∆ φ .…”

Positivity constraints on the low-energy constants of the chiral pion–nucleon Lagrangian

“…Indeed, there has recently been a renewed interest on the positivity issue. Following some modern lines, more stringent and precise upper bounds have been achieved on high-energy ππ cross section [21,22], with interesting applications of the Froissart-Martin upper bound to large-N C ππ scattering [23,24]. Similar crossing, unitarity and analyticity considerations could be applied here to extract improved upper bounds for the high-energy π N cross section.…”